Theorem lt2addrd 29516

Description: If the right-hand side of a 'less than' relationship is an addition, then we can express the left-hand side as an addition, too, where each term is respectively less than each term of the original right side. (Contributed by Thierry Arnoux, 15-Mar-2017.)

Hypotheses

Ref	Expression
lt2addrd.1	|- ( ph -> A e. RR )
lt2addrd.2	|- ( ph -> B e. RR )
lt2addrd.3	|- ( ph -> C e. RR )
lt2addrd.4	|- ( ph -> A < ( B + C ) )

Assertion

Ref	Expression
lt2addrd	|- ( ph -> E. b e. RR E. c e. RR ( A = ( b + c ) /\ b < B /\ c < C ) )

Distinct variable groups:   b, c, A   B, b, c   C, b, c

Allowed substitution hints:   ph( b, c)

Proof of Theorem lt2addrd

Step	Hyp	Ref	Expression
1		lt2addrd.2	. . 3 |- ( ph -> B e. RR )
2		lt2addrd.3	. . . . . 6 |- ( ph -> C e. RR )
3	1, 2	readdcld 10069	. . . . 5 |- ( ph -> ( B + C ) e. RR )
4		lt2addrd.1	. . . . 5 |- ( ph -> A e. RR )
5	3, 4	resubcld 10458	. . . 4 |- ( ph -> ( ( B + C ) - A ) e. RR )
6	5	rehalfcld 11279	. . 3 |- ( ph -> ( ( ( B + C ) - A ) / 2 ) e. RR )
7	1, 6	resubcld 10458	. 2 |- ( ph -> ( B - ( ( ( B + C ) - A ) / 2 ) ) e. RR )
8	2, 6	resubcld 10458	. 2 |- ( ph -> ( C - ( ( ( B + C ) - A ) / 2 ) ) e. RR )
9	2	recnd 10068	. . . . . 6 |- ( ph -> C e. CC )
10	1	recnd 10068	. . . . . . . . 9 |- ( ph -> B e. CC )
11	10, 9	addcld 10059	. . . . . . . 8 |- ( ph -> ( B + C ) e. CC )
12	4	recnd 10068	. . . . . . . 8 |- ( ph -> A e. CC )
13	11, 12	subcld 10392	. . . . . . 7 |- ( ph -> ( ( B + C ) - A ) e. CC )
14	13	halfcld 11277	. . . . . 6 |- ( ph -> ( ( ( B + C ) - A ) / 2 ) e. CC )
15	9, 14, 14	subsub4d 10423	. . . . 5 |- ( ph -> ( ( C - ( ( ( B + C ) - A ) / 2 ) ) - ( ( ( B + C ) - A ) / 2 ) ) = ( C - ( ( ( ( B + C ) - A ) / 2 ) + ( ( ( B + C ) - A ) / 2 ) ) ) )
16	15	oveq2d 6666	. . . 4 |- ( ph -> ( B + ( ( C - ( ( ( B + C ) - A ) / 2 ) ) - ( ( ( B + C ) - A ) / 2 ) ) ) = ( B + ( C - ( ( ( ( B + C ) - A ) / 2 ) + ( ( ( B + C ) - A ) / 2 ) ) ) ) )
17	9, 14	subcld 10392	. . . . 5 |- ( ph -> ( C - ( ( ( B + C ) - A ) / 2 ) ) e. CC )
18	10, 14, 17	subadd23d 10414	. . . 4 |- ( ph -> ( ( B - ( ( ( B + C ) - A ) / 2 ) ) + ( C - ( ( ( B + C ) - A ) / 2 ) ) ) = ( B + ( ( C - ( ( ( B + C ) - A ) / 2 ) ) - ( ( ( B + C ) - A ) / 2 ) ) ) )
19	13	2halvesd 11278	. . . . . 6 |- ( ph -> ( ( ( ( B + C ) - A ) / 2 ) + ( ( ( B + C ) - A ) / 2 ) ) = ( ( B + C ) - A ) )
20	19, 13	eqeltrd 2701	. . . . 5 |- ( ph -> ( ( ( ( B + C ) - A ) / 2 ) + ( ( ( B + C ) - A ) / 2 ) ) e. CC )
21	10, 9, 20	addsubassd 10412	. . . 4 |- ( ph -> ( ( B + C ) - ( ( ( ( B + C ) - A ) / 2 ) + ( ( ( B + C ) - A ) / 2 ) ) ) = ( B + ( C - ( ( ( ( B + C ) - A ) / 2 ) + ( ( ( B + C ) - A ) / 2 ) ) ) ) )
22	16, 18, 21	3eqtr4d 2666	. . 3 |- ( ph -> ( ( B - ( ( ( B + C ) - A ) / 2 ) ) + ( C - ( ( ( B + C ) - A ) / 2 ) ) ) = ( ( B + C ) - ( ( ( ( B + C ) - A ) / 2 ) + ( ( ( B + C ) - A ) / 2 ) ) ) )
23	19	oveq2d 6666	. . 3 |- ( ph -> ( ( B + C ) - ( ( ( ( B + C ) - A ) / 2 ) + ( ( ( B + C ) - A ) / 2 ) ) ) = ( ( B + C ) - ( ( B + C ) - A ) ) )
24	11, 12	nncand 10397	. . 3 |- ( ph -> ( ( B + C ) - ( ( B + C ) - A ) ) = A )
25	22, 23, 24	3eqtrrd 2661	. 2 |- ( ph -> A = ( ( B - ( ( ( B + C ) - A ) / 2 ) ) + ( C - ( ( ( B + C ) - A ) / 2 ) ) ) )
26		lt2addrd.4	. . . . 5 |- ( ph -> A < ( B + C ) )
27		difrp 11868	. . . . . 6 |- ( ( A e. RR /\ ( B + C ) e. RR ) -> ( A < ( B + C ) <-> ( ( B + C ) - A ) e. RR+ ) )
28	4, 3, 27	syl2anc 693	. . . . 5 |- ( ph -> ( A < ( B + C ) <-> ( ( B + C ) - A ) e. RR+ ) )
29	26, 28	mpbid 222	. . . 4 |- ( ph -> ( ( B + C ) - A ) e. RR+ )
30	29	rphalfcld 11884	. . 3 |- ( ph -> ( ( ( B + C ) - A ) / 2 ) e. RR+ )
31	1, 30	ltsubrpd 11904	. 2 |- ( ph -> ( B - ( ( ( B + C ) - A ) / 2 ) ) < B )
32	2, 30	ltsubrpd 11904	. 2 |- ( ph -> ( C - ( ( ( B + C ) - A ) / 2 ) ) < C )
33		oveq1 6657	. . . . 5 |- ( b = ( B - ( ( ( B + C ) - A ) / 2 ) ) -> ( b + c ) = ( ( B - ( ( ( B + C ) - A ) / 2 ) ) + c ) )
34	33	eqeq2d 2632	. . . 4 |- ( b = ( B - ( ( ( B + C ) - A ) / 2 ) ) -> ( A = ( b + c ) <-> A = ( ( B - ( ( ( B + C ) - A ) / 2 ) ) + c ) ) )
35		breq1 4656	. . . 4 |- ( b = ( B - ( ( ( B + C ) - A ) / 2 ) ) -> ( b < B <-> ( B - ( ( ( B + C ) - A ) / 2 ) ) < B ) )
36	34, 35	3anbi12d 1400	. . 3 |- ( b = ( B - ( ( ( B + C ) - A ) / 2 ) ) -> ( ( A = ( b + c ) /\ b < B /\ c < C ) <-> ( A = ( ( B - ( ( ( B + C ) - A ) / 2 ) ) + c ) /\ ( B - ( ( ( B + C ) - A ) / 2 ) ) < B /\ c < C ) ) )
37		oveq2 6658	. . . . 5 |- ( c = ( C - ( ( ( B + C ) - A ) / 2 ) ) -> ( ( B - ( ( ( B + C ) - A ) / 2 ) ) + c ) = ( ( B - ( ( ( B + C ) - A ) / 2 ) ) + ( C - ( ( ( B + C ) - A ) / 2 ) ) ) )
38	37	eqeq2d 2632	. . . 4 |- ( c = ( C - ( ( ( B + C ) - A ) / 2 ) ) -> ( A = ( ( B - ( ( ( B + C ) - A ) / 2 ) ) + c ) <-> A = ( ( B - ( ( ( B + C ) - A ) / 2 ) ) + ( C - ( ( ( B + C ) - A ) / 2 ) ) ) ) )
39		breq1 4656	. . . 4 |- ( c = ( C - ( ( ( B + C ) - A ) / 2 ) ) -> ( c < C <-> ( C - ( ( ( B + C ) - A ) / 2 ) ) < C ) )
40	38, 39	3anbi13d 1401	. . 3 |- ( c = ( C - ( ( ( B + C ) - A ) / 2 ) ) -> ( ( A = ( ( B - ( ( ( B + C ) - A ) / 2 ) ) + c ) /\ ( B - ( ( ( B + C ) - A ) / 2 ) ) < B /\ c < C ) <-> ( A = ( ( B - ( ( ( B + C ) - A ) / 2 ) ) + ( C - ( ( ( B + C ) - A ) / 2 ) ) ) /\ ( B - ( ( ( B + C ) - A ) / 2 ) ) < B /\ ( C - ( ( ( B + C ) - A ) / 2 ) ) < C ) ) )
41	36, 40	rspc2ev 3324	. 2 |- ( ( ( B - ( ( ( B + C ) - A ) / 2 ) ) e. RR /\ ( C - ( ( ( B + C ) - A ) / 2 ) ) e. RR /\ ( A = ( ( B - ( ( ( B + C ) - A ) / 2 ) ) + ( C - ( ( ( B + C ) - A ) / 2 ) ) ) /\ ( B - ( ( ( B + C ) - A ) / 2 ) ) < B /\ ( C - ( ( ( B + C ) - A ) / 2 ) ) < C ) ) -> E. b e. RR E. c e. RR ( A = ( b + c ) /\ b < B /\ c < C ) )
42	7, 8, 25, 31, 32, 41	syl113anc 1338	1 |- ( ph -> E. b e. RR E. c e. RR ( A = ( b + c ) /\ b < B /\ c < C ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   class class class wbr 4653  (class class class)co 6650   CCcc 9934   RRcr 9935   + caddc 9939   < clt 10074   - cmin 10266   / cdiv 10684   2c2 11070   RR+crp 11832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-2 11079  df-rp 11833

This theorem is referenced by:  xlt2addrd  29523